Q: How do you calculate 6/2(1+2) or 48/2(9+3)? What’s the deal with this orders of operation business?

Mathematician: Now that the Physicist and I have answered questions like these ones at least 9 times via email, I figure we should get this horrible topic out of the way once and for all with a short post.

The “order of operations” tells us the sequence in which we should carry out mathematical operations. It is merely a matter of convention (not a matter of right or wrong), giving us a standardized way to decide whether 6/2*3 is (6/2)*3 or 6/(2*3). The convention has been accepted just so that when different people see an equation they can agree on its meaning, even if they hate each other’s guts. You could come up with a different convention, but why bother? Let other people waste their time coming up with arbitrary conventions so that you can merely waste time reading posts about them.

The rules to follow are:

1. Things in parenthesis get computed before things not in parenthesis (this is because of the (little known fact) that parentheses kick ass).

3. Next, multiplication and division get carried out from left to right. So a*b/c*d is ((a*b)/c)*d. Note that division can be thought of as still carrying out a multiplication since a/b = a*(1/b), so it makes a certain sense that neither division nor multiplication takes precedence over one another.

4. Finally, additions and subtractions get carried out from left to right. Subtraction can be thought of as being like addition since a – b = a + (-b) so it makes sense that subtraction is treated on an equal footing with addition.

People sometimes use the expressions PEMDAS, “Please Excuse My Dear Aunt Sally” or “Puke Exhaust Mud Dolls Autumn Sasquatch” to remember these rules. These all stand for “Parenthesis, Exponents, Multiplication and Division, Addition and Subtraction”. It is easy to get confused by these mnemonics though because they don’t capture the fact that multiplication is treated the same way as division (carried out left to right), and addition is treated the same way as subtraction (carried out left to right). That’s why I prefer: “Puke Exhaust Mud And Dolls Autumn And Sasquatch”

One interesting thing to note about the order of operations is that there is nothing interesting about them whatsoever. But that implies there is something interesting about them. This is a contradiction, and hence the order of operations form a paradox. They are also about as fun as getting to watch a fish crawl up the stairs as a reward for answering math questions.

But, that won’t stop me from giving an example. Consider:

a^b*c/d+e-f.

Without a convention, it has many possible interpretations, such as

a^(b*(c/(d+e)))-f

or

(a^b)*((c/(d+e))-f)

which could be horribly confusing for people trying to communicate with each other. But, the correct interpretation according to the rules mentioned above, is:

(((((a^b)*c)/d)+e)-f).

When in doubt about what the orders of operation are for an expression, there is a simple solution which works even better than tattooing PEMDAS to your face. Just paste the expression into the google search bar and hit “search”. For instance, if I paste in:

4^2*3/6+1-5

it gives back

(((4^2) * 3) / 6) + 1 – 5 = 4.

Not only is this the correct answer, but it shows you explicitely the order of operations that were used, so it’s useful for learning.

Google even gets this one right:

4^2*3/6 + 1 – 5/14*3 + 6/10 – 4*2/14*6^3/18 + 14

which is the kind of question little sisters give you when they are old enough to finally sort of get what a mathematician is.

It’s important to note that not all calculators handle the order of operations in the standard way. Why? Because people who make calculators hate children. The preceding sentence was not intended to be a factual statement.

Also, a lot of programming languages use the standard order of operation rules, but then you have exceptions like Smalltalk.

237 Responses to Q: How do you calculate 6/2(1+2) or 48/2(9+3)? What’s the deal with this orders of operation business?

Dear Kate first of all i dont care for what my calculator says, a incorrectly programmed.machine doesn’t have to be the smarter one just because it says so! Its a computer and computers are operated, if you operate them correctly ,they will give you.the correct answer to this equation.

In order to let your calc see the correlation between the multiplier “2” over the brackets, you need to let it prioritize that part firsts.. If not it will follow pemdas and give you a invitee answer 9!

Let me explain why this answer is in correct and maybe you’ll understand it afterwards..

Lets get clear that the concept of multiplication is nothing else than short writing a certain amount of the same you have. For example if I have 2+2+2 I have 3 times the unit “2” , right!?

So the multiplier becomes 3. To write this in a multiplication, that could be 3*2 in other words and also in literal speaking you pronounce this as 3times 2..so far so good right?

OK the same concept apply if you replace the value of the unit..for example with an letter “b”.. If I say you have 2b that means literally you have 2 times whatever “b”stands for. Perfectly equal to saying 2b is exactly b+b!

Ok ..now let’s explain that in this case writing down 2b there is a unwritten multiplication symbol between 2 and “b”, that is OK as it eases things up, but the correlation of 2 over “b” remains!

Now if we get back to the equation 6÷2(1+2),

You see the 2 standing directly before the parentheses, even though in many case it would be fine to place the * symbol yourself, now its not! Why…
Because if you do so, you’ll ignore the fact of why the symbol is left out.. The 6 and the 2 are separate units as the are split by the “÷” symbol, the “2” is not separated and should not be from whatever value is between the parentheses.

Your order of operations ( pemdas) rules, tell you to solve parentheses first. But as the correct way of dealing with the parentheses would be to see or rewrite it as
2 units of (1+2), like in 2b or 4a etc..

That can be rewritten as ((1+2)+(1+2)) or as ((2*(1+2)), on its own you could do 2*(1+2), but not now, since its part in a equation!
If you do, your pemdas will neglect the correlation and true meaning over what multiplication is! what the multiplier is over which unit..

So solving the equation as 6÷2* (1+2) would be incorrect as in this case it is absolutely not the same as in doing 6÷((2*(1+2))!

I asked both my husband and my 15 year old son and both said 1. Thank god. I was about to evict everyone out of this house. The calculator is wrong and so is the internet to come up with 9. Of course if you have a real calculator (scientific or graphing) that can handle an equation, that will spit out 1 all day long. The 2 belongs to the (1+2) portion of the equation. 6/2(3) The parentheses don’t just disappear! You still have to resolve it before you can solve! I’m 41 years old taking pre-calc. Undoing a life time of misunderstanding about math.

Also since many of you seem to forget you can’t have a coefficient without a variable. Let’s check our work just to be sure.
6/2(x)=9 or 6/2(x)=1 if we solve for x it must equal 3
6/2(x)=9
6(x)=18
X= 3
6/2(x)=1
6(x)=1 ITS DEFINITELY NOT ONE PEOPLE. I DONT UNDERSTAND WHY PEOPLE WANT TO PUSH IMPLIED MULTIPLICATION. ITS NOT A THING IT NEVER WAS. THIS ISN’T ALGEBRA ITS FOURTH GRADE ARITHMETIC. YOU DO WHATS INSIDE THE PARENTHESES FIRST AND THEN YOU MOVE ON. ITS 9 .. so says my scientific calculator, a graphing calculator, and Google. Tighten up folks

We can beg to differ on this one..but 6÷2(1+2) is not the same as 6/2(1+2) when you enter it into a calculator or into google. The calculator is trying to solve 6 OVER 2 (or 6 HALVES like a fraction.) and handles the problem left to right. It doesn’t understand “multiplication by juxtaposition” rules or that a(b+c) is the same as ab+ac.

The Casio fx115ES Plus Natural is the best calculator you will ever buy. It allows you to enter an equation exactly as stated, even a fraction with the numerator over the denominator.

I didn’t need the calculator to solve this.
I read it as 6 divided by 2 (2+1) also. I do not have a division sign on my phone. Multiplication and division are done from left to right. Even you distributed the answer is 9 people. You forget, seriously
6/2(2)+6/2(1) = 3(2)+ 3(1) = 6+3=9

None of these things should be needed to solve this. This is grade school math, it isn’t even algebra. You simply follow the order of operations. INSIDE the parentheses first, exponents, multiply or divide left to right and add or subtract left to right. Those have always been the rules of math and still are.

“multiplication denoted by juxtaposition is interpreted as having higher precedence than division, so that 1/2x equals 1/(2x), not (1/2)x. For example, the manuscript submission instructions for the Physical Review journals state that multiplication is of higher precedence than division with a slash,[9] and this is also the convention observed in prominent physics textbooks such as the Course of Theoretical Physics by Landau and Lifshitz and the Feynman Lectures on Physics.[nb 1]”

Aksa, any book may announce its convention. The problem here is that some people assume that without convention, the “standard” or “default” is 1/2x = (1/2)x because 2x = 2 * x

I will post later with all the math texts I found that ALL use 1/2x = 1/(2x) and NONE use 1/2x = 1/2 * x.
They explain what quantities are and factor groups as well. Basic stuff. Sure 2x = 2 * x, but 2x is a product of factors before it is used in the order of operations. in the order of operations, we need to know first what the operators and operands are. In this case, 2x is a single quantity, that is, it is one operand.

Does anyone what to argue this fact? If so, provide a reference please.

Yes. 1/2(x) is literally one half multiplied by x . I spoke to multiple professors about this and they say neither of us our correct because the problem is ambiguous how it is written. Type the problem into any scientific calculator, Google or wolphram alpha because they all have the same answer as me.

None of you are making sense. U can’t prove what you are saying
6/2(3)=9
Let’s substitute (3) with (x) so I can prove it
6/2(x)=9 multiply both sides by 2
6(x)=18
X=3 . U can’t do it if the answer is 1. The guy way above is correct it’s different when there are variables and constants.

The problem is that people do not finish the parentheses step before jumping to the multiplication/division step. To begin with, I am going to put the equation into its simplest terms.

you have six apples and two sets of people, each consisting of 1 male and 2 females. How many apples will each individual get? If you have six apples and six people, each person gets one apple, not nine apples.

That is why it is critical to understand that writing 2 (2 +1) is not the same as 2 x (2 +1). The first would be solved in the “parentheses” (“bracket” for the Brits) step. The second, with the inclusion of the multiplication function would fall under the “multiplication/division” step.

People are making the mistake of identifying the function of multiplying as arbitrary, so you solve from left to right. It is not arbitrary, as putting a 2 next to the parentheses brings distributive properties into play (meaning the constant or variable next to the parentheses defines what happens within the parentheses). So it is still part of the “parentheses” step.

You are not at the “multiplication/division” step, yet. It is when 2 x (2 +1) is shown, that it becomes part of the “multiplication/division” step.

Like spelling and punctuation matter in sentence structure, how you write the equation matters. If you start incorporating “slang” into math, you get an equation of constants coming up with two solutions, in this case 1 and 9.

Before someone burns their diploma, for thinking that 6/2(1+2)=9, I thought I would show the difference between 2(1+2) and 2 x (1+2), especially as it relates to this equation.

As was shown, in a prior post, with apples, the equation written reflects the process of having apples and people, with the solution being how many apples each person gets.

An equation 6 ÷ 2 x (1+2) would show an equation where you are looking at calculating two different solutions, to get to a final solution. For example, you have 6 apples. Each apple makes (1 cookie and 2 muffins). If you divide the apples between 2 people, how many total cookies and muffins can each individual make.

So, unlike the first equation (where you are just dealing with apples and people), in this equation, you are dealing with apples, people and baked goods. And the associative property would apply, as this would be solved in the multiplication/division step.

You could figure out total apples to total baked goods and divide by 2 people 6 x (1+2) ÷ 2 = 9

Or total apples per individual, then number of baked goods based per apple each individual has 6 ÷ 2 x (1+2) = 9

However, saying that 6 ÷ 2(1+2) and 6 ÷ 2 x (1+2) are equations saying the same thing, is wrong.

p.s. since my father has his doctorate in math … I should see what answer he comes up with (whether 1 or 9) … lol

It’s not a matter of one answer being right and the other wrong. The mathematical notation itself is ambiguous. The correct answer will differ depending on how one chooses to interpret the expression. Some will see it as (a): (6/2)(1+2), in which case the “correct” answer is 9. Others will read it as (b): 6/(2(1+2)), which will mean the “correct” answer is 1. To avoid this type of ambiguity many academic mathematical authorities have established BY CONVENTION that “multiplication by juxtaposition” or “implied multiplication”, as in (b:) 2(1+2) above, should be performed before division, making the “correct” answer 1.

I’m not here ro discuss if one answer is better or not than the other or if the convention showed here are wrong or not but to tell the author have forgotten one little thing.

Writing convention for the sign of the operation differ from a place to an other in the world.

here an exemple.

As far as I know in the USA (and maybe in other countries too) 6/2 say two things “6 dived by 2” and “6 over 2” at the same time. I think its creating a lot of confusion in child and adult mind. While for me (and many other ppl around the world) I would write 6:2. or 6÷2 for “6 divided by 2” and 6/2 for “6 over 2”.

I think that’s why there is so much argumentation over these two problems.

Le me show you with 6/2(1+2) :

-> With USA model : the result of “6 dived by 2” that I multiply by the result of “1 plus 2”.

-> In the other way of writin : 6 over the result of “2 multiplied by the result of 1 plus 2”

You will ask me wut why didn’t you read “6 over 2” ? because I have also learnt (in more advanced math class) that the absence of multiplying sign in 2(1+2) imply a particular relation between 2 and (1+2).

But be very carefull its not because here they gave me the same result it always will be the case! Especialy when you work with variables, resolving the equation with the first method can lead to awfull mistakes.

I hope I was clear enought for every one who search answers on this subject.

The problem with Ted and Ray is they made up their minds about the answer then weakly defended their argument for ego’s sake. Caper gave 6 valid examples as well as countered Ray and Teds one example. Scientifically speaking, all 6 of Capers examples must be proven false in order prove him wrong. I graduated with a math degree and agree with capers arguments. Sue correctly argued the parentheses must be resolved first thus 2(3) takes priority over 6/2. In short unless Ray or Ted can imperially disprove all 6 of capers arguments then caper (who disproved their sole argument) has won the debate.

To get a historical analysis and explanation, you need to read all my previous posts, including the stuff about operands. Everyone knows how to interpret the operators, but you need to really read what is an operand, and what are coefficients, and how to interpret their meaning. Notice how I quote actual books and there hasn’t been a single legitimate rebuttal of the book references. Not a single book in history, to my knowledge, has ever written 6/2x with the intended meaning 6/2*x.
This refutes the argument for “since 2x = 2 * x, we rewrite 6/2x = 6 / 2 * x”

2x is the PRODUCT of 2 and x. This is a single value, or “quantity” as understood by algebra texts and authors. Quantities are to be identified before operations are executed on those quantities.

The fact that people are arguing about this really shows how far our educational system has fallen. No, this is not an “ambiguous” statement. Parsing is not an exercise in hand-waving and interpretation and opinions about convention; it’s a rigorously studied problem in formal languages and computer science. You have the grammar and the parsing rules, and you parse according to them. If you accept PEMDAS, then the answer is 9. END. OF. STORY. It doesn’t matter whether Einstein himself wrote 1/2x as shorthand when he meant 1/(2x). He’s wrong, and you’re wrong if you’re trying to refute the rules of a formal language with an appeal to authority.

If you want to say that ‘/’ is really a viniculum and thus a grouping operator of the same precedence as parentheses, or if you want to add an additional precedence level to PEMDAS for implied multiplication, that’s fine. But you’re changing the rules to get to what you feel intuitively is the way things *should* be.

6 is a single quantity. If you show factors, the quantity is precisely the same and obviously cannot change: 2(3). If we show 3 using addends, 3=2+1, we have 2(2+1).

A group of coefficients are coefficients of each other. “Xyz” for example. X is a coefficient of yz. And vice versa, etc. Also, the coefficients can be rearranged in any order. We use sequential as convention. Xyz = yzx; etc.

Distribute yes please.

But I see what you did there. You introduced something new when people won’t acknowledge 6/2x = 6 ÷ 2x = 3/x

I got back to a ÷ a.
We know the coefficients of “a”. It’s 1. It’s there. We just don’t write it. But we CAN.
a ÷ a = 1a ÷ 1a = 1

This is not the original problem. The problem shown on both twitter and instagram doesn’t have a fraction bar, a but an actual obelus. Therefore, the answer saying it is worked 6/(2(1+2)) is wrong. 6÷2(1+2) is the correct equation.
It is worked:
6÷2(1+2)
6÷2(3)
3(3)
6

I am confused by the fraction bar obelus comment. I had understood that there’s no difference. To be clear:
6 ÷ 2(1+2)
6 ÷ (2+4) by distribution.
6 ÷ 6
1
Does a fraction bar return a different answer than ÷ ?

“doesn’t have a fraction bar, a but an actual obelus. Therefore, the answer saying it is worked 6/(2(1+2)) is wrong. 6÷2(1+2) is the correct equation.
It is worked:
6÷2(1+2)
6÷2(3)
3(3)
6”

So, I know this seems redundant, however,
6 ÷ 2(1+2)
y ÷ x(a+b)
y ÷ (xa + xb)
6 ÷ (2 + 4) = 1 by distribution, and per 1st and 2nd response from this site.
Is this an improper reduction?
And are ÷, / fundamentally the same?

The 3rd response below, contradicts the 2 earlier responses.

In 2 previous responses from this site:

“…..The “grammar”, which dates back to the 1800’s im various texts, says 1/2x is “one over the product of 2 and x”, that is, 1/2x = 1/(2x).

Not a single published algebra text in history teaches what you are projecting. …”
________________________
And the 2nd response:

“….. 6 is a single quantity. If you show factors, the quantity is precisely the same and obviously cannot change: 2(3). If we show 3 using addends, 3=2+1, we have 2(2+1).

A group of coefficients are coefficients of each other. “Xyz” for example. X is a coefficient of yz. And vice versa, etc. Also, the coefficients can be rearranged in any order.

Distribute yes please.

…. I got back to a ÷ a. We know the coefficients of “a”. It’s 1. It’s there. We just don’t write it. But we CAN.

a ÷ a = 1a ÷ 1a = 1 use sequential as convention. Xyz = yzx; etc. …”
__________________________
So I’m confuse by this 3rd response, the end value of 6:

“… This is not the original problem. The problem shown on both twitter and instagram doesn’t have a fraction bar, a but an actual obelus. Therefore, the answer saying it is worked 6/(2(1+2)) is wrong. 6÷2(1+2) is the correct equation.
It is worked:
6÷2(1+2)
6÷2(3)
3(3)
6 ….”

Here’s the thing. When you google this it spits out (6/2)*(1+2) This equation is NOT the same as 6/2(1+2) Which is why google gets you 9. by the way is incorrect. This equation in fact has two terms 6 which is divided by the other term 2(1+2). The way google reads this is like stated earlier is as follows – first term (6/2) multipled by the second term (1+2). Math is a language and just like English just be interpreted correctly. There and they’re – are the same word with two different meaning. You have to understand grammar to know which is correct in its own context. Math is the same way. You just can’t assume what is right. You can not separate 2 from (1+2). It IS a term its self and until it is solved you can not continue the equation.

As everyone knows, Excel and Google will give an answer of 9 for 6/2(1+2).
This is, as people have pointed out, because Excel and Google calculate the 6/2 first, and then calculate the rest as 3(1+2). It then calculates using the order of operations, thus 3(1+2) = 3(3) = 9.

However, if you simply use order of operations, and apply the idea that a(b+c) is really a*(b+c), and also ignore any stupid rules about solidi being different to obeli, then 6/2(1+2) should be calculated by calculating the parentheses first, which then brings you to: 6/2(3). You THEN, calculate the division, from left to right.
6 / 2 = 3, 3 x 3 = 9.

I would like to change my answer to N.A. Reason being is because this equation is not written in a mathematical notation. The ONLY way to answer this is to ASSUME what is supposed to be written as to what you perceive is written. Here’s a grammatical example: Three plus for minus two and one too. There are so many error and it just doesn’t make sense. If we TRY to solve this using logic there WILL be a multitude of assumed and preceived solutions. Ambiguous is a word you will see ALOT when trying to understand this equation. Simply put is that this equation is incorrect from the start and cannot be solved.

I guess it’s a shame that the math of quite recent textbooks was successfully taught. And airplanes, Saturn rockets, atomic reactors, cars, electronics, were all invented and developed with the perfectly adequate and well understood math. But now that same math is cited as not proper math, no longer correct, in need of revision.

Another generation and all that math becomes hieroglyphic symbols. I guess source documents will have solutions that only used to be right.

Matthew Collinson.
There actually IS a rule that states 6/2x = 3/x ; not 3x
It is in algebra books.
Books even go so far as to define quantities (operands), as well as what coefficients are, as well as factors and products of factors. Every single one of these principles shows that 6/2x = 3/x
Simply let x=3 now.